1988 ap calculus ab multiple choice answers questions

4 plus 2 times minus 2, that'sĤ minus 4, that's 0. Right? Because then you'd have minusĢ, plus 2 times this, and they'd cancel out. Rid of this guy? Well let me replace row 3 with So 1 minus 0 is 1, 2 minus 0 isĢ, 1 minus 1 is 0, 1 minus minus 2, that's 1 plusĢ, which is 3. This one here? Well I can subtract- I can Which are always going to have the coefficient 1, or the entryġ, it should be the only non-zero term in my row. Here, because I want to get into reduced row echelon form. My equals sign, or the augmented part of the matrix. Same this time, so I get a 0, 0, 1, minus 2, and essentially Get rid of this minus 2 term right there.

Now what can we do? Well, let's see if we can And then 4 minus 2 timesĨ, is minus 16, 4 minus 16 is minus 12. So we get 2 minus 2 times 1 isĠ, 4 minus 2 times 2, well that's 0, 0 minus 2 timesġ, that's minus 2. The third equation minus two times our first equation. This guy right here, let's replace our third equation with There you go, that looks good soįar, it looks like column, or x2 which is represented byĬolumn two, looks like it might be a free variable, but Is 0, 2 minus 1 is 1, 1- negative 1 minus 1 is minus 2,Īnd then 12 minus 8 is 4. Subtract- let me replace the second row with the second So what can we do? I'm going to keep my first row I want to zero out these two rows right here. Now let's put this guy into reduced row echelon form. Coefficients on the x4 areġ, minus 1, and 6. Than variables, so I probably won't be able to constrain Least my gut feeling says, look, I have fewer equations Systems of linear equations using augmented matrices, at I talked about reduced row echelon form, and solving Pivot numbers are just the first non-zero entry in a row, significant to put matrices into the appropriate form (for example Reduced Row Echelon form).Įquations of four unknowns. In two dimensions, for example, constants (X, Y) represent a point, but in there dimensions it would be a line, with infinite points on the Z-plane (X, Y, aZ). That being said, it depends on your solution and the dimensions you're working in what geometric representation is appropriate.

Reduced Row Echelon Form just results form elementary row operations (ie, performing equivalent operations, that do not change overall value) until you have rows like 'X +0Y = a & 0X + Y = b'Ĭoncerning points, lines, planes, etc., this is generally only brought up for intuition's sake during early stages of matrix algebra, as it can get difficult to comprehend when you're working in higher dimensions. Thus the 'n', the last column representing the 'solution' will change when performing row operations, but the value of the equations doesn't shave, as you're performing equal operations on both (all) sides of the =.

Augmented forms of matrices have the 'solution' (x+ y = n) IN it, usually represented as the last column, or an Ax1 matrix following the original matrix.

1988 ap calculus ab multiple choice answers questions